APPROXIMATION OF LIPSCHITZ FUNCTION BY (E,q) (C,1) SUMMABILITY METHOD

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1 e-joural of Sciece & Techology (e-jst) APPRXIMATIN F LIPSCHITZ FUNCTIN BY (E,q) (C,) SUMMABILITY METHD Shyam Lal ad Ramasharay Sigh yadav. Professor, Faculty of sciece, Departmet of Mathematics Baaras Hidu Uiversity, Varaasi (Idia) shyam_lal@rediffmail.com. Research Assistat, Departmet of Mathematics, Harcourt Butler Techical Uiversity, Kapur (Idia) sigh.344@rediffmail.com Abstract I the paper a ew theorem o the degree of approximatio of Lipschitz fuctio o by (c,) meas is established Keywords ad phrases: Degree of approximatio, Fourier series, (E,q) summability. 5 Mathematics subject classificatio primary 4B5, 4B8. Itroductio First time i 99 Lal [ ] has bee discussed the degree of approximatio by (c,) (E,) product summability meas. The degree of approximatio of fuctio belogig to geeralized Lipschitz class by Nörlud meas has bee determied by several researchers of moder aalysis. But othig seems to have bee doe so for i the directio of study of degree of approximatio of Lipα fuctio by product summability meas of the form (E,q) (c,). I preset paper, the degree of approximatio of Lipschitz fuctio by (E,q) (c,) meas of its Fourier series has bee determied. Defiitio ad Notatios Let f(t) be periodic with period π ad itergrable i the sese of Lebesgue. The Fourier series of f(t) is give by f ( t ) a ( cos si ) (.) a t b t A fuctio f Lip if f ( x t) f ( x) ( t ) for 53

2 e-joural of Sciece & Techology (e-jst) The degree of approximatio of a fuctio f:r R by a trigoometric polyomial t order is defied by, zygmud (968,,p-4.). t f sup t( x) f ( x) : x R Let u be a ifiite series with the sequece of partial sum s. If s s as The TheSeries u issaid tosummable tos by C, method. For q>, theeuler mea isgive by Hardy q E ( q) q s ad the methodis deoted as( E, q) q If E s, The u issaid tosummable tos by( E, q) method. ( E, q) trasformatio of i.e( E, q)( c,) trasformof s is defied by qc, E q ( q) o qc, if E s as The, wesay that u issaid to besummable tos by( E, q) c method. Itisdeoted by u s ( E, q) c Weshall use thefollowig otatio ( t) f ( x t) f ( x t) f ( x) qc, si ( ) t E q ( q) ( ) si t (4), 7 54

3 e-joural of Sciece & Techology (e-jst) 3. Mai Theorem The degree of approximatio of f Lip has bee determied i the followig form: Theorem: If f: R R is π- periodic ad Legesgue itegrable o [ π, π] ad f Lip the the degree of approximatio of f by (E,q)(c,) product summability meas of its Fourier series is give by, qc, ( ) E ( x ) f ( x ) log( ) e, ( ) qc, where E ( x) q s ( q) is( E, q)( c,) measof fourier series (.). 4. Lemmas For the proof of our theorem the followig Lemmas are required. Lemma 4.:- For t Proof:- + c, q si ( ) t Eq () t ( q) si t c, q si ( ) t Eq () t ( q) si t q ( ) si t ( q) si t q ( ) ( q) ( ) q ( ) ( q) 55

4 e-joural of Sciece & Techology (e-jst) Lemma 4.:- For t,, by Joda's Lemma. ( ) t t si c, q si ( ) t Eq () t ( ) ( q) si t si ( ) t q ( ) ( q) si t q ( ) ( q) t q ( ) t t 5. Proof of the mai theorem Let s (x) deote th partial sum of series (.) at t=x the followig Tichmarsh (p.4 &43). s ( C,) mea of s( x). ( ) t si f ( x) f ( x t) f ( x t) f ( x) tdt ( ) si t ( ) t si f ( x) ( t) dt ( ) si t (4), 7 56

5 e-joural of Sciece & Techology (e-jst) Now, q f ( x) ( ) q Let us cosider I at first ( ) t si q ( q) ( ) si t ( ) t si q ( q) ( ) si t q, c t() () () I ( ) q, c t () ( ) ( ) ( t) dt bylemma (4.) ( ) ( ) o( t ) dt ( t) Lip ( ) ( ) ( ) 57

6 e-joural of Sciece & Techology (e-jst) ( ) ( ) Let uscosider I. q, I c t() ( t) dt by lemma(4.) ( ) t t, log( ), (4), 7 58

7 e-joural of Sciece & Techology (e-jst), log( ), log Now, I I,, log( ), ( ) log( ) e,,, This is complete the proof of the mai theorem. ACKNWLEDGMENT Authors are grateful to Dr.L.M.Tripathi Ex-Proffessor & Head, Departmet of Maths, Baaras Hidu Uiversity, Varaasi for suggestig the problem ad the Dr. B. Rai. Ex. Head Departmet of maths, Uiversity of Allahabad, Allahabad 59

8 e-joural of Sciece & Techology (e-jst) ad also Dr. S.U. Siddiqui Head Departmet of maths, HBTI. Kapur, who has tae the pai the see the mauscript. Refereces. G. Das, Sagita Mishra.&B.K Ray (997): Degree of Approximatio by Euler meas of r th Derived Fourier series i the HÖldermetric. j. Nat. Acad. Mathvol.,8-. Shyam Lal (999): The degree of approximatio of cojugats of almost Lipschitz fuctios by (c,)(e,) meas, Bul.Cal. Maths. Soc., 9,(), Shyam Lal (999): degree of approximatio of fuctios belogig to the weighted[l p (t)] by (c,)(e,) meas. Taag Joural of Maths, 3, K.Qureshi (98): o degree of approximatio of a fuctio belogig to the class Lip, Idia joural of pure applied Mathematics. 3, K.Qureshi (98): o degree of approximatio of a fuctio belogig to the weighted [L p (t)] class, Idia Joural & Pure & Applied Mathematics 3, L.Mc Fodde (94): Absolute NÖrlud summability Due maths J.9, E.C. Titchmarsh (99): Theory of fuctio, 3, A.Zygmud: Trigeometric series, d Rev. - cambrige Uiv. Press. Cambrige 5, 968. (4), 7 6